Consider another integer \(b\ge 2\)  and its corresponding symbolic space \(\Sigma _b^+\) , let \(\Phi _{\mathbf {I}_b}:\Sigma _b^+\rightarrow [0,1]\)  be its associated canonical mapping. Let \(\nu ^+\)  be a \(\sigma \) -invariant probability measure on \(\Sigma _b^+\) . Let \(\Pi \)  be the family of orthogonal projections from \(\mathbb {R}^2\)  to its 1-dimensional subspaces. Hochman and Shmerkin [1]} proved that if \(\log a/\log b\)  is irrational, then
\(\underline{\dim }_H \pi (\Phi _{\mathbf {I}_a}(\mu ^+)\times \Phi _{\mathbf {I}_b}(\nu ^+))=\min \lbrace 1,\underline{\dim }_H \Phi _{\mathbf {I}_a}(\mu ^+)\times \Phi _{\mathbf {I}_b}(\nu ^+)\rbrace \) 
